We will study which sets defined by placing conditions on the statistical behaviour of orbits are Borel
(assuming that \(X\)  is compact if necessary). Recall that \((X,T)\)  is a Polish dynamical system and
\((f_n)\)  is a sequence of bounded continuous real-valued functions on \(X\) 
which generate the weak\(^*\)  topology on \(\operatorname{\mathcal {M}}(X)\)  (cf. Theorem 17.19 of [1]}).
That is, \(\mu _k \rightarrow \mu \)  as \(k\rightarrow \infty \)  if and only if for each \(n\)  we have \(\int f_n\, d\mu _k \rightarrow \int f_n\, d\mu \)  as \(k\rightarrow \infty \) .
Recall also that \(A_k\varphi \)  stands for the \(k\) -th ergodic average of \(\varphi \)  along the orbit of \(y\) , see (REF ).
